Integrand size = 21, antiderivative size = 257 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (c^2 d f-e g\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}+\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e^2 \left (c^2 d^2-e^2\right )^{5/2}} \] Output:
1/6*b*c*(-d*g+e*f)*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)/(e*x+d)^2+1/2*b*c*(c ^2*d*f-e*g)*(-c^2*x^2+1)^(1/2)/(c^2*d^2-e^2)^2/(e*x+d)-1/3*(-d*g+e*f)*(a+b *arcsin(c*x))/e^2/(e*x+d)^3-1/2*g*(a+b*arcsin(c*x))/e^2/(e*x+d)^2+1/6*b*c^ 3*(e^2*(-4*d*g+e*f)+c^2*d^2*(d*g+2*e*f))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^ (1/2)/(-c^2*x^2+1)^(1/2))/e^2/(c^2*d^2-e^2)^(5/2)
Time = 0.45 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.25 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\frac {\frac {a (-2 e f+2 d g)}{(d+e x)^3}-\frac {3 a g}{(d+e x)^2}+\frac {b c e \sqrt {1-c^2 x^2} \left (c^2 d \left (4 d e f-d^2 g+3 e^2 f x\right )-e^2 (2 d g+e (f+3 g x))\right )}{\left (-c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {b (2 e f+d g+3 e g x) \arcsin (c x)}{(d+e x)^3}+\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right ) \log (d+e x)}{(-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}-\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{(-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}}{6 e^2} \] Input:
Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]
Output:
((a*(-2*e*f + 2*d*g))/(d + e*x)^3 - (3*a*g)/(d + e*x)^2 + (b*c*e*Sqrt[1 - c^2*x^2]*(c^2*d*(4*d*e*f - d^2*g + 3*e^2*f*x) - e^2*(2*d*g + e*(f + 3*g*x) )))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (b*(2*e*f + d*g + 3*e*g*x)*ArcSin [c*x])/(d + e*x)^3 + (b*c^3*(e^2*(e*f - 4*d*g) + c^2*d^2*(2*e*f + d*g))*Lo g[d + e*x])/((-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (b*c^3*( e^2*(e*f - 4*d*g) + c^2*d^2*(2*e*f + d*g))*Log[e + c^2*d*x + Sqrt[-(c^2*d^ 2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]))/(6*e^2)
Time = 0.56 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5252, 27, 688, 27, 679, 488, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx\) |
\(\Big \downarrow \) 5252 |
\(\displaystyle -b c \int -\frac {2 e f+d g+3 e g x}{6 e^2 (d+e x)^3 \sqrt {1-c^2 x^2}}dx-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b c \int \frac {2 e f+d g+3 e g x}{(d+e x)^3 \sqrt {1-c^2 x^2}}dx}{6 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}\) |
\(\Big \downarrow \) 688 |
\(\displaystyle \frac {b c \left (\frac {\int -\frac {2 \left (-d (2 e f+d g) c^2+e (e f-d g) x c^2+3 e^2 g\right )}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 \left (c^2 d^2-e^2\right )}+\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^2}\right )}{6 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\int \frac {-d (2 e f+d g) c^2+e (e f-d g) x c^2+3 e^2 g}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}\right )}{6 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}\) |
\(\Big \downarrow \) 679 |
\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {-\frac {c^2 \left (c^2 d^2 (d g+2 e f)+e^2 (e f-4 d g)\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}-\frac {3 e^2 \sqrt {1-c^2 x^2} \left (c^2 d f-e g\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}}{c^2 d^2-e^2}\right )}{6 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\frac {c^2 \left (c^2 d^2 (d g+2 e f)+e^2 (e f-4 d g)\right ) \int \frac {1}{-c^2 d^2+e^2-\frac {\left (d x c^2+e\right )^2}{1-c^2 x^2}}d\frac {d x c^2+e}{\sqrt {1-c^2 x^2}}}{c^2 d^2-e^2}-\frac {3 e^2 \sqrt {1-c^2 x^2} \left (c^2 d f-e g\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}}{c^2 d^2-e^2}\right )}{6 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}+\frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {-\frac {c^2 \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (c^2 d^2 (d g+2 e f)+e^2 (e f-4 d g)\right )}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {3 e^2 \sqrt {1-c^2 x^2} \left (c^2 d f-e g\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}}{c^2 d^2-e^2}\right )}{6 e^2}\) |
Input:
Int[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]
Output:
-1/3*((e*f - d*g)*(a + b*ArcSin[c*x]))/(e^2*(d + e*x)^3) - (g*(a + b*ArcSi n[c*x]))/(2*e^2*(d + e*x)^2) + (b*c*((e*(e*f - d*g)*Sqrt[1 - c^2*x^2])/((c ^2*d^2 - e^2)*(d + e*x)^2) - ((-3*e^2*(c^2*d*f - e*g)*Sqrt[1 - c^2*x^2])/( (c^2*d^2 - e^2)*(d + e*x)) - (c^2*(e^2*(e*f - 4*d*g) + c^2*d^2*(2*e*f + d* g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^ 2 - e^2)^(3/2))/(c^2*d^2 - e^2)))/(6*e^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_ Symbol] :> With[{u = IntHide[Px*(d + e*x)^m, x]}, Simp[(a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
Leaf count of result is larger than twice the leaf count of optimal. \(907\) vs. \(2(237)=474\).
Time = 0.38 (sec) , antiderivative size = 908, normalized size of antiderivative = 3.53
method | result | size |
parts | \(a \left (-\frac {g}{2 e^{2} \left (e x +d \right )^{2}}-\frac {-d g +e f}{3 e^{2} \left (e x +d \right )^{3}}\right )+\frac {b \left (\frac {c^{4} \arcsin \left (c x \right ) d g}{3 e^{2} \left (c x e +c d \right )^{3}}-\frac {c^{4} \arcsin \left (c x \right ) f}{3 e \left (c x e +c d \right )^{3}}-\frac {c^{3} \arcsin \left (c x \right ) g}{2 e^{2} \left (c x e +c d \right )^{2}}+\frac {c^{3} \left (\frac {3 g \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}-\frac {2 c \left (d g -e f \right ) \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {3 d c e \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{2 \left (c^{2} d^{2}-e^{2}\right )}+\frac {e^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{3}}\right )}{6 e^{2}}\right )}{c}\) | \(908\) |
derivativedivides | \(\frac {a \,c^{3} \left (-\frac {g}{2 e^{2} \left (c x e +c d \right )^{2}}+\frac {c \left (d g -e f \right )}{3 e^{2} \left (c x e +c d \right )^{3}}\right )+b \,c^{3} \left (-\frac {\arcsin \left (c x \right ) g}{2 e^{2} \left (c x e +c d \right )^{2}}+\frac {\arcsin \left (c x \right ) c d g}{3 e^{2} \left (c x e +c d \right )^{3}}-\frac {\arcsin \left (c x \right ) c f}{3 e \left (c x e +c d \right )^{3}}+\frac {\frac {3 g \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}-\frac {2 c \left (d g -e f \right ) \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {3 d c e \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{2 \left (c^{2} d^{2}-e^{2}\right )}+\frac {e^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{3}}}{6 e^{2}}\right )}{c}\) | \(912\) |
default | \(\frac {a \,c^{3} \left (-\frac {g}{2 e^{2} \left (c x e +c d \right )^{2}}+\frac {c \left (d g -e f \right )}{3 e^{2} \left (c x e +c d \right )^{3}}\right )+b \,c^{3} \left (-\frac {\arcsin \left (c x \right ) g}{2 e^{2} \left (c x e +c d \right )^{2}}+\frac {\arcsin \left (c x \right ) c d g}{3 e^{2} \left (c x e +c d \right )^{3}}-\frac {\arcsin \left (c x \right ) c f}{3 e \left (c x e +c d \right )^{3}}+\frac {\frac {3 g \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}-\frac {2 c \left (d g -e f \right ) \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {3 d c e \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{2 \left (c^{2} d^{2}-e^{2}\right )}+\frac {e^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{3}}}{6 e^{2}}\right )}{c}\) | \(912\) |
Input:
int((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^4,x,method=_RETURNVERBOSE)
Output:
a*(-1/2*g/e^2/(e*x+d)^2-1/3*(-d*g+e*f)/e^2/(e*x+d)^3)+b/c*(1/3*c^4*arcsin( c*x)/e^2/(c*e*x+c*d)^3*d*g-1/3*c^4*arcsin(c*x)/e/(c*e*x+c*d)^3*f-1/2*c^3*a rcsin(c*x)*g/e^2/(c*e*x+c*d)^2+1/6*c^3/e^2*(3*g/e^2*(1/(c^2*d^2-e^2)*e^2/( c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)-d* c*e/(c^2*d^2-e^2)/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d* c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+ d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e)))-2*c*(d*g-e*f)/e^3*(1/2/(c^2 *d^2-e^2)*e^2/(c*x+d*c/e)^2*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e ^2)/e^2)^(1/2)+3/2*d*c*e/(c^2*d^2-e^2)*(1/(c^2*d^2-e^2)*e^2/(c*x+d*c/e)*(- (c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)-d*c*e/(c^2*d^2- e^2)/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/ e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d ^2-e^2)/e^2)^(1/2))/(c*x+d*c/e)))+1/2/(c^2*d^2-e^2)*e^2/(-(c^2*d^2-e^2)/e^ 2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^ 2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c* x+d*c/e)))))
Leaf count of result is larger than twice the leaf count of optimal. 946 vs. \(2 (237) = 474\).
Time = 16.93 (sec) , antiderivative size = 1920, normalized size of antiderivative = 7.47 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:
integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^4,x, algorithm="fricas")
Output:
[-1/12*(6*(a*c^6*d^6*e - 3*a*c^4*d^4*e^3 + 3*a*c^2*d^2*e^5 - a*e^7)*g*x - sqrt(-c^2*d^2 + e^2)*(((2*b*c^5*d^2*e^4 + b*c^3*e^6)*f + (b*c^5*d^3*e^3 - 4*b*c^3*d*e^5)*g)*x^3 + 3*((2*b*c^5*d^3*e^3 + b*c^3*d*e^5)*f + (b*c^5*d^4* e^2 - 4*b*c^3*d^2*e^4)*g)*x^2 + (2*b*c^5*d^5*e + b*c^3*d^3*e^3)*f + (b*c^5 *d^6 - 4*b*c^3*d^4*e^2)*g + 3*((2*b*c^5*d^4*e^2 + b*c^3*d^2*e^4)*f + (b*c^ 5*d^5*e - 4*b*c^3*d^3*e^3)*g)*x)*log((2*c^2*d*e*x - c^2*d^2 + (2*c^4*d^2 - c^2*e^2)*x^2 + 2*sqrt(-c^2*d^2 + e^2)*(c^2*d*x + e)*sqrt(-c^2*x^2 + 1) + 2*e^2)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(a*c^6*d^6*e - 3*a*c^4*d^4*e^3 + 3*a *c^2*d^2*e^5 - a*e^7)*f + 2*(a*c^6*d^7 - 3*a*c^4*d^5*e^2 + 3*a*c^2*d^3*e^4 - a*d*e^6)*g + 2*(3*(b*c^6*d^6*e - 3*b*c^4*d^4*e^3 + 3*b*c^2*d^2*e^5 - b* e^7)*g*x + 2*(b*c^6*d^6*e - 3*b*c^4*d^4*e^3 + 3*b*c^2*d^2*e^5 - b*e^7)*f + (b*c^6*d^7 - 3*b*c^4*d^5*e^2 + 3*b*c^2*d^3*e^4 - b*d*e^6)*g)*arcsin(c*x) - 2*sqrt(-c^2*x^2 + 1)*(3*((b*c^5*d^3*e^4 - b*c^3*d*e^6)*f - (b*c^3*d^2*e^ 5 - b*c*e^7)*g)*x^2 + (4*b*c^5*d^5*e^2 - 5*b*c^3*d^3*e^4 + b*c*d*e^6)*f - (b*c^5*d^6*e + b*c^3*d^4*e^3 - 2*b*c*d^2*e^5)*g + ((7*b*c^5*d^4*e^3 - 8*b* c^3*d^2*e^5 + b*c*e^7)*f - (b*c^5*d^5*e^2 + 4*b*c^3*d^3*e^4 - 5*b*c*d*e^6) *g)*x))/(c^6*d^9*e^2 - 3*c^4*d^7*e^4 + 3*c^2*d^5*e^6 - d^3*e^8 + (c^6*d^6* e^5 - 3*c^4*d^4*e^7 + 3*c^2*d^2*e^9 - e^11)*x^3 + 3*(c^6*d^7*e^4 - 3*c^4*d ^5*e^6 + 3*c^2*d^3*e^8 - d*e^10)*x^2 + 3*(c^6*d^8*e^3 - 3*c^4*d^6*e^5 + 3* c^2*d^4*e^7 - d^2*e^9)*x), -1/6*(3*(a*c^6*d^6*e - 3*a*c^4*d^4*e^3 + 3*a...
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (d + e x\right )^{4}}\, dx \] Input:
integrate((g*x+f)*(a+b*asin(c*x))/(e*x+d)**4,x)
Output:
Integral((a + b*asin(c*x))*(f + g*x)/(d + e*x)**4, x)
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{4}} \,d x } \] Input:
integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^4,x, algorithm="maxima")
Output:
-1/6*(3*e*x + d)*a*g/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2) - 1/3 *a*f/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 1/6*((3*b*e*g*x + 2*b *e*f + b*d*g)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 6*(e^5*x^3 + 3* d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2)*integrate(1/6*(3*b*c*e*g*x + 2*b*c*e*f + b*c*d*g)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1))/(c^4*e^5*x^7 + 3*c^4*d *e^4*x^6 - 3*c^2*d^2*e^3*x^3 - c^2*d^3*e^2*x^2 + (3*c^4*d^2*e^3 - c^2*e^5) *x^5 + (c^4*d^3*e^2 - 3*c^2*d*e^4)*x^4 + (c^2*e^5*x^5 + 3*c^2*d*e^4*x^4 - 3*d^2*e^3*x - d^3*e^2 + (3*c^2*d^2*e^3 - e^5)*x^3 + (c^2*d^3*e^2 - 3*d*e^4 )*x^2)*e^(log(c*x + 1) + log(-c*x + 1))), x))/(e^5*x^3 + 3*d*e^4*x^2 + 3*d ^2*e^3*x + d^3*e^2)
Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^4,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d+e\,x\right )}^4} \,d x \] Input:
int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^4,x)
Output:
int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^4, x)
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\frac {6 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b \,d^{3} e^{2} f +18 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b \,d^{2} e^{3} f x +18 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b d \,e^{4} f \,x^{2}+6 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b \,e^{5} f \,x^{3}+6 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b \,d^{3} e^{2} g +18 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b \,d^{2} e^{3} g x +18 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b d \,e^{4} g \,x^{2}+6 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b \,e^{5} g \,x^{3}-a d g -2 a e f -3 a e g x}{6 e^{2} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:
int((g*x+f)*(a+b*asin(c*x))/(e*x+d)^4,x)
Output:
(6*int(asin(c*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e **4*x**4),x)*b*d**3*e**2*f + 18*int(asin(c*x)/(d**4 + 4*d**3*e*x + 6*d**2* e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4),x)*b*d**2*e**3*f*x + 18*int(asin(c* x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4),x)*b *d*e**4*f*x**2 + 6*int(asin(c*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4 *d*e**3*x**3 + e**4*x**4),x)*b*e**5*f*x**3 + 6*int((asin(c*x)*x)/(d**4 + 4 *d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4),x)*b*d**3*e**2*g + 18*int((asin(c*x)*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x **3 + e**4*x**4),x)*b*d**2*e**3*g*x + 18*int((asin(c*x)*x)/(d**4 + 4*d**3* e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4),x)*b*d*e**4*g*x**2 + 6 *int((asin(c*x)*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4),x)*b*e**5*g*x**3 - a*d*g - 2*a*e*f - 3*a*e*g*x)/(6*e**2*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))